You find asteroids in our solar system earth by taking photos of the same point of the sky at different times. The stars far away will not move so any point of light that has moved is close. With multiple observation and the using earth moment around the sun, you determine the distance and it orbit.
The first large asteroid was discovered just like that in the 19th century but without photographs so you had to do draw what you observed on paper. Ceres that is the larges asteroid was found that way in 1801. Not a lot was discovered this way
Later you get photographic plates and then film more was discovered.
The majority of asteroids in our solar system has been discovered recently with digital cameras, robotic telescopes, and software that analyze the data
So you systematically take images of the sky and compare them over time.
One of the programs can be found [https://cneos.jpl.nasa.gov/about/search_program.html](https://cneos.jpl.nasa.gov/about/search_program.html)
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